The Science of Sticky Spheres

Take a dozen marbles, all the same size, and squeeze them into a compact, three-dimensional cluster. Now count the number of points where the marbles touch one another. What is the maximum number of contact points you can possibly achieve with 12 marbles? What geometric arrangement yields this greatest contact number? Is the optimal cluster unique, or are there multiple solutions that all give the same maximum?

When I first heard these questions asked, they did not seem overly challenging. For a cluster consisting of two, three, four or five equal-size spheres, I was pretty sure I knew the answers. But I soon learned that the problem gets harder in a hurry as the size of the cluster increases. Over the past three years, the maximum contact number has been determined for clusters of up to 11 spheres. Finding those answers required a variety of mathematical tools drawn from graph theory and geometry, as well as extensive computations and, at a few crucial junctures, building ball-and-stick models with a set of toys called Geomags. For clusters of 12 or more spheres, the answers remain unknown.

To be stumped by such simple questions about small clumps of spheres is humbling—but perhaps not too surprising. Sphere-packing problems are notoriously tricky. Some of them have resisted analysis for centuries.